Calculus Examples

$(10 t^{- 3} + 12 t^{- 9} + 4 t^{3}) d t$

Step 1

Write $(10 t^{- 3} + 12 t^{- 9} + 4 t^{3}) d t$ as a function.

$f (t) = (10 t^{- 3} + 12 t^{- 9} + 4 t^{3}) d t$

Step 2

The function $F (t)$ can be found by finding the indefinite integral of the derivative $f (t)$ .

$F (t) = \int f (t) d t$

Step 3

Set up the integral to solve.

$F (t) = \int (10 t^{- 3} + 12 t^{- 9} + 4 t^{3}) d t d t$

Step 4

Since $d$ is constant with respect to $t$ , move $d$ out of the integral.

$d \int (10 t^{- 3} + 12 t^{- 9} + 4 t^{3}) t d t$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$d \int (10 t^{- 3} + 12 t^{- 9}) t + 4 t^{3} t d t$

Step 5.2

Apply the distributive property.

$d \int 10 t^{- 3} t + 12 t^{- 9} t + 4 t^{3} t d t$

Step 5.3

Raise $t$ to the power of $1$ .

$d \int 10 (t^{- 3} t^{1}) + 12 t^{- 9} t + 4 t^{3} t d t$

Step 5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$d \int 10 t^{- 3 + 1} + 12 t^{- 9} t + 4 t^{3} t d t$

Step 5.5

Add $- 3$ and $1$ .

$d \int 10 t^{- 2} + 12 t^{- 9} t + 4 t^{3} t d t$

Step 5.6

Raise $t$ to the power of $1$ .

$d \int 10 t^{- 2} + 12 (t^{- 9} t^{1}) + 4 t^{3} t d t$

Step 5.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$d \int 10 t^{- 2} + 12 t^{- 9 + 1} + 4 t^{3} t d t$

Step 5.8

Add $- 9$ and $1$ .

$d \int 10 t^{- 2} + 12 t^{- 8} + 4 t^{3} t d t$

Step 5.9

Raise $t$ to the power of $1$ .

$d \int 10 t^{- 2} + 12 t^{- 8} + 4 (t^{3} t^{1}) d t$

Step 5.10

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$d \int 10 t^{- 2} + 12 t^{- 8} + 4 t^{3 + 1} d t$

Step 5.11

Add $3$ and $1$ .

$d \int 10 t^{- 2} + 12 t^{- 8} + 4 t^{4} d t$

Step 5.12

Move $12 t^{- 8}$ .

$d \int 10 t^{- 2} + 4 t^{4} + 12 t^{- 8} d t$

Step 5.13

Reorder $10 t^{- 2}$ and $4 t^{4}$ .

$d \int 4 t^{4} + 10 t^{- 2} + 12 t^{- 8} d t$

Step 6

Split the single integral into multiple integrals.

$d (\int 4 t^{4} d t + \int 10 t^{- 2} d t + \int 12 t^{- 8} d t)$

Step 7

Since $4$ is constant with respect to $t$ , move $4$ out of the integral.

$d (4 \int t^{4} d t + \int 10 t^{- 2} d t + \int 12 t^{- 8} d t)$

Step 8

By the Power Rule, the integral of $t^{4}$ with respect to $t$ is $\frac{1}{5} t^{5}$ .

$d (4 (\frac{1}{5} t^{5} + C) + \int 10 t^{- 2} d t + \int 12 t^{- 8} d t)$

Step 9

Since $10$ is constant with respect to $t$ , move $10$ out of the integral.

$d (4 (\frac{1}{5} t^{5} + C) + 10 \int t^{- 2} d t + \int 12 t^{- 8} d t)$

Step 10

By the Power Rule, the integral of $t^{- 2}$ with respect to $t$ is $- t^{- 1}$ .

$d (4 (\frac{1}{5} t^{5} + C) + 10 (- t^{- 1} + C) + \int 12 t^{- 8} d t)$

Step 11

Since $12$ is constant with respect to $t$ , move $12$ out of the integral.

$d (4 (\frac{1}{5} t^{5} + C) + 10 (- t^{- 1} + C) + 12 \int t^{- 8} d t)$

Step 12

By the Power Rule, the integral of $t^{- 8}$ with respect to $t$ is $- \frac{1}{7} t^{- 7}$ .

$d (4 (\frac{1}{5} t^{5} + C) + 10 (- t^{- 1} + C) + 12 (- \frac{1}{7} t^{- 7} + C))$

Step 13

Simplify.

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Step 13.1

Simplify.

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Step 13.1.1

Combine $\frac{1}{5}$ and $t^{5}$ .

$d (4 (\frac{t^{5}}{5} + C) + 10 (- t^{- 1} + C) + 12 (- \frac{1}{7} t^{- 7} + C))$

Step 13.1.2

Combine $t^{- 7}$ and $\frac{1}{7}$ .

$d (4 (\frac{t^{5}}{5} + C) + 10 (- t^{- 1} + C) + 12 (- \frac{t^{- 7}}{7} + C))$

Step 13.1.3

Move $t^{- 7}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$d (4 (\frac{t^{5}}{5} + C) + 10 (- t^{- 1} + C) + 12 (- \frac{1}{7 t^{7}} + C))$

Step 13.2

Simplify.

$d (\frac{4 t^{5}}{5} - \frac{10}{t} - \frac{12}{7 t^{7}}) + C$

Step 13.3

Reorder terms.

$d (\frac{4}{5} t^{5} - \frac{10}{t} - \frac{12}{7 t^{7}}) + C$

Step 14

The answer is the antiderivative of the function $f (t) = (10 t^{- 3} + 12 t^{- 9} + 4 t^{3}) d t$ .

$F (t) =$ $d (\frac{4}{5} t^{5} - \frac{10}{t} - \frac{12}{7 t^{7}}) + C$